data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Simplify $\\sqrt{7\\cdot 2}\\cdot \\sqrt{2^3\\cdot 7^3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "196",
"style": "rule"
} | {
"index": "7500",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), point \\(P\\) is located one-third of the way along segment \\(AB\\) closer to point \\(A\\). Point \\(R\\) is one-third of the way along segment \\(PB\\) closer to point \\(P\\), and point \\(Q\\) lies on segment \\(BC\\) such that angles \\(PCB\\) and \\(RQB\\) are congrue... | {
"ground_truth": "9:2",
"style": "rule"
} | {
"index": "7501",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "7502",
"split": "train"
} |
math | [
{
"content": "Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "7503",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be positive real numbers. Find the maximum value of\n\\[2(a - x)(x + \\sqrt{x^2 + b^2})\\]in terms of $a$ and $b.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "a^2 + b^2",
"style": "rule"
} | {
"index": "7504",
"split": "train"
} |
math | [
{
"content": "Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "7505",
"split": "train"
} |
math | [
{
"content": "The area of the plane region enclosed by the two curves $y=x^2$, $y=\\frac{1}{4}x^2$ and the line $y=1$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "7506",
"split": "train"
} |
math | [
{
"content": "A labor employment service center has 7 volunteers ready to provide labor employment guidance on the streets over the weekend, with 6 people to be arranged for duty on Saturday and Sunday. If 3 people are arranged for each day, there are a total of _______ different arrangements. (Answer with a nu... | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "7507",
"split": "train"
} |
math | [
{
"content": "Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1, a,$ and $b$ or $\\tfrac{1}{b}, \\tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "7508",
"split": "train"
} |
math | [
{
"content": "Let \\( G \\) be the centroid of \\(\\triangle ABC\\). Given \\( BG \\perp CG \\) and \\( BC = \\sqrt{2} \\), find the maximum value of \\( AB + AC \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "7509",
"split": "train"
} |
math | [
{
"content": "The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. Find the smallest possible difference between the two 2-digit numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "7510",
"split": "train"
} |
math | [
{
"content": "What is the absolute value of the difference between the squares of 101 and 99? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "7511",
"split": "train"
} |
math | [
{
"content": "There are 2019 bags, each bag containing 2019 beads with a total weight of 1 kilogram. The beads in each bag are numbered \\(1, 2, \\cdots, 2019\\). A selection is considered proper if and only if beads with different numbers are selected from different bags (at most one bead from each bag), and t... | {
"ground_truth": "2018!",
"style": "rule"
} | {
"index": "7512",
"split": "train"
} |
math | [
{
"content": "In triangle $A B C$ with altitude $A D, \\angle B A C=45^{\\circ}, D B=3$, and $C D=2$. Find the area of triangle $A B C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "7513",
"split": "train"
} |
math | [
{
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day.... | {
"ground_truth": "\\frac{349}{500}",
"style": "rule"
} | {
"index": "7514",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be positive real numbers such that $x^{2}+y^{2}=1$ and \\left(3 x-4 x^{3}\\right)\\left(3 y-4 y^{3}\\right)=-\\frac{1}{2}$. Compute $x+y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "7515",
"split": "train"
} |
math | [
{
"content": "Given the vectors $\\overrightarrow{a}=(\\cos 40^{\\circ},\\sin 40^{\\circ})$, $\\overrightarrow{b}=(\\sin 20^{\\circ},\\cos 20^{\\circ})$, and $\\overrightarrow{u}= \\sqrt {3} \\overrightarrow{a}+λ \\overrightarrow{b}$ (where $λ∈R$), find the minimum value of $|\\overrightarrow{u}|$. Please outpu... | {
"ground_truth": "\\frac {\\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "7516",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than $900$ are either a perfect cube or a perfect square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "7517",
"split": "train"
} |
math | [
{
"content": "The graph of the function $y=f(x)$ is shown below. For all $x > 4$, it is true that $f(x) > 0.4$. If $f(x) = \\frac{x^2}{Ax^2 + Bx + C}$, where $A,B,$ and $C$ are integers, then find $A+B+C$. [asy]\nimport graph; size(10.9cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen... | {
"ground_truth": "-12",
"style": "rule"
} | {
"index": "7518",
"split": "train"
} |
math | [
{
"content": "In the \"Black White Pair\" game, a common game among children often used to determine who goes first, participants (three or more) reveal their hands simultaneously, using the palm (white) or the back of the hand (black) to decide the winner. If one person shows a gesture different from everyone ... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "7519",
"split": "train"
} |
math | [
{
"content": "Olga Ivanovna, the homeroom teacher of class 5B, is staging a \"Mathematical Ballet\". She wants to arrange the boys and girls so that every girl has exactly 2 boys at a distance of 5 meters from her. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7520",
"split": "train"
} |
math | [
{
"content": "How many $6$ -digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3003",
"style": "rule"
} | {
"index": "7521",
"split": "train"
} |
math | [
{
"content": "In a household, when someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.3, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability of the phone call being answered within the first four rings. Please outpu... | {
"ground_truth": "0.9",
"style": "rule"
} | {
"index": "7522",
"split": "train"
} |
math | [
{
"content": "A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$ . How many distinct $3$ -letter words can be made from a subset of the letters $c, o, m, b, o$ , where each letter in the list is used no more than the number of times it appears? Please output the final answer with... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "7523",
"split": "train"
} |
math | [
{
"content": "The average of a set of distinct primes is 27. What is the largest prime that can be in this set? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "139",
"style": "rule"
} | {
"index": "7524",
"split": "train"
} |
math | [
{
"content": "A line with slope $k$ intersects the parabola $y^{2}=4x$ at points $A$ and $B$, and is tangent to the circle $(x-5)^{2}+y^{2}=9$ at point $M$, where $M$ is the midpoint of segment $AB$. Find the value of $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "7525",
"split": "train"
} |
math | [
{
"content": "Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.\n\na) Write down any five-digit palindromic number that is divisible by 5.\n\nb) How many five-digit palindromic numbers are there that are divisible by 5? Ple... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "7526",
"split": "train"
} |
math | [
{
"content": "An equivalent of the expression\n$\\left(\\frac{x^2+1}{x}\\right)\\left(\\frac{y^2+1}{y}\\right)+\\left(\\frac{x^2-1}{y}\\right)\\left(\\frac{y^2-1}{x}\\right)$, $xy \\not= 0$,\nis: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2xy+\\frac{2}{xy}",
"style": "rule"
} | {
"index": "7527",
"split": "train"
} |
math | [
{
"content": "Given a point $P$ on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}$, $F_{2}$, find the eccentricity of the ellipse given that $\\overrightarrow{PF_{1}} \\cdot \\overrightarrow{PF_{2}}=0$ and $\\tan \\angle PF_{1}F_{2}= \\frac{1}{2}$. Please output the fina... | {
"ground_truth": "\\frac{\\sqrt{5}}{3}",
"style": "rule"
} | {
"index": "7528",
"split": "train"
} |
math | [
{
"content": "There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\\times 3 \\times 3$ cube with these $27$ cubes. What ... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "7529",
"split": "train"
} |
math | [
{
"content": "Find the largest real number $c$ such that $$ \\sum_{i=1}^{n}\\sum_{j=1}^{n}(n-|i-j|)x_ix_j \\geq c\\sum_{j=1}^{n}x^2_i $$ for any positive integer $n $ and any real numbers $x_1,x_2,\\dots,x_n.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7530",
"split": "train"
} |
math | [
{
"content": "Given that point $P(x,y)$ is a moving point on the circle $x^{2}+y^{2}=2y$,\n(1) Find the range of $z=2x+y$;\n(2) If $x+y+a\\geqslant 0$ always holds, find the range of real numbers $a$;\n(3) Find the maximum and minimum values of $x^{2}+y^{2}-16x+4y$. Please output the final answer within \\boxed... | {
"ground_truth": "6-2\\sqrt{73}",
"style": "rule"
} | {
"index": "7531",
"split": "train"
} |
math | [
{
"content": "There is a game called \"Twenty-Four Points\" with the following rules: Choose four natural numbers between $1$ and $13$, and perform addition, subtraction, multiplication, and division operations on these four numbers (each number can be used only once and parentheses can be used) to make the res... | {
"ground_truth": "3 \\times \\left(10 - 6 + 4\\right)",
"style": "rule"
} | {
"index": "7532",
"split": "train"
} |
math | [
{
"content": "If Heidi can paint a wall in 45 minutes, what fractional part of the wall can she paint in 9 minutes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "7533",
"split": "train"
} |
math | [
{
"content": "How many integers \\(n\\) satisfy \\((n+5)(n-9) \\le 0\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "7534",
"split": "train"
} |
math | [
{
"content": "Find the area bounded by the graph of $ y\\equal{}\\sqrt[3]{x\\plus{}\\sqrt{x^2\\plus{}1}}\\plus{}\\sqrt[3]{x\\minus{}\\sqrt{x^2\\plus{}1}}$ , the line $ x\\minus{}y\\minus{}1\\equal{}0$ and the $ x$ axis. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5/8",
"style": "rule"
} | {
"index": "7535",
"split": "train"
} |
math | [
{
"content": "What is the sum of the greatest common factor of 3 and 6 and the least common multiple of 3 and 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "7536",
"split": "train"
} |
math | [
{
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3, -1]$, $f(x) = -(x+2)^2$, and when $x \\in [-1, 3)$, $f(x) = x$. Calculate the value of $f(1) + f(2) + f(3) + \\ldots + f(2015)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "7537",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a regular tetrahedron and Let $E$ be a point inside the face $ABC.$ Denote by $s$ the sum of the distances from $E$ to the faces $DAB, DBC, DCA,$ and by $S$ the sum of the distances from $E$ to the edges $AB, BC, CA.$ Then $\\frac{s}{S}$ equals Please output the final answer within \... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "7538",
"split": "train"
} |
math | [
{
"content": "Fill in the table with the numbers $0, 1, 2, \\cdots, 14, 15$ so that for each row and each column, the remainders when divided by 4 are exactly $0, 1, 2, 3$ each, and the quotients when divided by 4 are also exactly $0, 1, 2, 3$ each, and determine the product of the four numbers in the bottom ro... | {
"ground_truth": "32760",
"style": "rule"
} | {
"index": "7539",
"split": "train"
} |
math | [
{
"content": "Let $p,$ $q,$ $r,$ $s$ be real numbers such that\n\\[\\frac{(p - q)(r - s)}{(q - r)(s - p)} = \\frac{3}{7}.\\]Find the sum of all possible values of\n\\[\\frac{(p - r)(q - s)}{(p - q)(r - s)}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{3}",
"style": "rule"
} | {
"index": "7540",
"split": "train"
} |
math | [
{
"content": "How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\\pi x$, the line $y=-0.1$ and the line $x=5.1?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "7541",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer \\( n \\) such that all the roots of \\( z^6 - z^3 + 1 = 0 \\) are \\( n^{\\text{th}} \\) roots of unity? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7542",
"split": "train"
} |
math | [
{
"content": "There are two values of $a$ for which the equation $4x^2+ax+8x+9=0$ has only one solution for $x$. What is the sum of those values of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "7543",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, where $XY = 5$, $YZ = 12$, $XZ = 13$, and $YM$ is the angle bisector from vertex $Y$. If $YM = m \\sqrt{2}$, find $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{60}{17}",
"style": "rule"
} | {
"index": "7544",
"split": "train"
} |
math | [
{
"content": "An urn contains 101 balls, exactly 3 of which are red. The balls are drawn one by one without replacement. On which draw is it most likely to pull the second red ball? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "7545",
"split": "train"
} |
math | [
{
"content": "Yannick picks a number $N$ randomly from the set of positive integers such that the probability that $n$ is selected is $2^{-n}$ for each positive integer $n$. He then puts $N$ identical slips of paper numbered 1 through $N$ into a hat and gives the hat to Annie. Annie does not know the value of $... | {
"ground_truth": "\\frac{1}{2 \\ln 2-1}",
"style": "rule"
} | {
"index": "7546",
"split": "train"
} |
math | [
{
"content": "The diagram below shows \\( \\triangle ABC \\), which is isosceles with \\( AB = AC \\) and \\( \\angle A = 20^\\circ \\). The point \\( D \\) lies on \\( AC \\) such that \\( AD = BC \\). The segment \\( BD \\) is constructed as shown. Determine \\( \\angle ABD \\) in degrees. Please output the f... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7547",
"split": "train"
} |
math | [
{
"content": "Simplify $\\sqrt{18} \\times \\sqrt{32} \\times \\sqrt{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24\\sqrt{2}",
"style": "rule"
} | {
"index": "7548",
"split": "train"
} |
math | [
{
"content": "The number $2^{1993}+3^{1993}$ is a multiple of $5.$ What is the units digit of the quotient $\\frac{2^{1993}+3^{1993}}{5}?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "7549",
"split": "train"
} |
math | [
{
"content": "Which number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down, rows go right and left.)\n\\[\\begin{tabular}{ccccc} 10 & 6 & 4 & 3 & 2 \\\\ 11 & 7 & 14 & 10 & 8 \\\\ 8 & 3 & 4 & 5 & 9 \\\\ 13 & 4 & 15 & 12 & 1 \\\\ 8 & 2 & 5 & 9 & 3 \\end{t... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7550",
"split": "train"
} |
math | [
{
"content": "Alice is making a batch of cookies and needs $2\\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7551",
"split": "train"
} |
math | [
{
"content": "April has four different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2880",
"style": "rule"
} | {
"index": "7552",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(x+1)^{42}$, what is the coefficient of the $x^2$ term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "861",
"style": "rule"
} | {
"index": "7553",
"split": "train"
} |
math | [
{
"content": "Among all the four-digit numbers without repeated digits, how many numbers have the digit in the thousandth place 2 greater than the digit in the unit place? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "448",
"style": "rule"
} | {
"index": "7554",
"split": "train"
} |
math | [
{
"content": "Given a rectangle \\(ABCD\\). On two sides of the rectangle, different points are chosen: six points on \\(AB\\) and seven points on \\(BC\\). How many different triangles can be formed with vertices at the chosen points? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "231",
"style": "rule"
} | {
"index": "7555",
"split": "train"
} |
math | [
{
"content": "A product originally priced at \\$120 receives a discount of 8%. Calculate the percentage increase needed to return the reduced price to its original amount. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8.7\\%",
"style": "rule"
} | {
"index": "7556",
"split": "train"
} |
math | [
{
"content": "Given the sequence \\(\\left\\{a_{n}\\right\\}\\), which satisfies\n\\[ \na_{1}=0,\\left|a_{n+1}\\right|=\\left|a_{n}-2\\right| \n\\]\n\nLet \\(S\\) be the sum of the first 2016 terms of the sequence \\(\\left\\{a_{n}\\right\\}\\). Determine the maximum value of \\(S\\). Please output the final an... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "7557",
"split": "train"
} |
math | [
{
"content": "Given a geometric progression \\( b_1, b_2, \\ldots, b_{3000} \\) with all positive terms and a total sum \\( S \\). It is known that if every term with an index that is a multiple of 3 (i.e., \\( b_3, b_6, \\ldots, b_{3000} \\)) is increased by 50 times, the sum \\( S \\) increases by 10 times. H... | {
"ground_truth": "\\frac{11}{8}",
"style": "rule"
} | {
"index": "7558",
"split": "train"
} |
math | [
{
"content": "When someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.2, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability that the phone call is answered within the first four rings. Please output the final answe... | {
"ground_truth": "0.8",
"style": "rule"
} | {
"index": "7559",
"split": "train"
} |
math | [
{
"content": "A right cylinder with a height of 8 inches is enclosed inside another cylindrical shell of the same height but with a radius 1 inch greater than the inner cylinder. The radius of the inner cylinder is 3 inches. What is the total surface area of the space between the two cylinders, in square inches... | {
"ground_truth": "16\\pi",
"style": "rule"
} | {
"index": "7560",
"split": "train"
} |
math | [
{
"content": "A weather forecast predicts that for each day from Monday to Friday this week, there is a $40\\%$ chance of sun, a $25\\%$ chance of getting 4 inches of rain, and a $35\\%$ chance of getting 10 inches of rain. What is the expected value of the total number of inches of rain which will fall from Mo... | {
"ground_truth": "22.5",
"style": "rule"
} | {
"index": "7561",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number $A$ that satisfies the following conditions:\n\na) Its notation ends with the digit 6;\n\nb) By moving the digit 6 from the end of the number to its beginning, the number increases fourfold. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "153846",
"style": "rule"
} | {
"index": "7562",
"split": "train"
} |
math | [
{
"content": "Kyle knows that $4 = 2^{5r+1}$. What is the value of $r$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "7563",
"split": "train"
} |
math | [
{
"content": "Each of the three aluminum cans holds an integer number of liters of water. The second can holds 1.5 times more than the first, and the volume of the third can is equal to \\(\\frac{4^3}{3}\\) times the volume of the first can. What is the total number of liters of water that the three cans togeth... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "7564",
"split": "train"
} |
math | [
{
"content": "For each positive integer $n$, let $n!$ denote the product $1\\cdot 2\\cdot 3\\cdot\\,\\cdots\\,\\cdot (n-1)\\cdot n$.\n\nWhat is the remainder when $9!$ is divided by $11$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7565",
"split": "train"
} |
math | [
{
"content": "A block of iron solidifies from molten iron, and its volume reduces by $\\frac{1}{34}$. Then, if this block of iron melts back into molten iron (with no loss in volume), by how much does its volume increase? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{33}",
"style": "rule"
} | {
"index": "7566",
"split": "train"
} |
math | [
{
"content": "The expression $12y^2-65y+42$ can be written as $(Ay-14)(By-3),$ where $A$ and $B$ are integers. What is $AB + A$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "7567",
"split": "train"
} |
math | [
{
"content": "Integers $x$ and $y$ with $x > y > 0$ satisfy $x + y + xy = 119$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "7568",
"split": "train"
} |
math | [
{
"content": "What is the greatest common divisor of $39$ and $91$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "7569",
"split": "train"
} |
math | [
{
"content": "Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$ . Find the maximal possible value of $mn$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2350",
"style": "rule"
} | {
"index": "7570",
"split": "train"
} |
math | [
{
"content": "Let $w,$ $x,$ $y,$ and $z$ be nonnegative numbers whose sum is 100. Find the largest possible value of\n\\[wx + xy + yz.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": "7571",
"split": "train"
} |
math | [
{
"content": "In a number line, point $P$ is at 3 and $V$ is at 33. The number line between 3 and 33 is divided into six equal parts by the points $Q, R, S, T, U$. What is the sum of the lengths of $PS$ and $TV$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "7572",
"split": "train"
} |
math | [
{
"content": "Find the minimum distance from a point on the curve y=e^{x} to the line y=x. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "7573",
"split": "train"
} |
math | [
{
"content": "Let $A$ be a point on the circle $x^2 + y^2 - 12x + 31 = 0,$ and let $B$ be a point on the parabola $y^2 = 4x.$ Find the smallest possible distance $AB.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "7574",
"split": "train"
} |
math | [
{
"content": "There are several solid and hollow circles arranged in a certain pattern as follows: ●○●●○●●●○●○●●○●●●… Among the first 2001 circles, find the number of hollow circles. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "667",
"style": "rule"
} | {
"index": "7575",
"split": "train"
} |
math | [
{
"content": "The number of solutions in positive integers of $2x+3y=763$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "7576",
"split": "train"
} |
math | [
{
"content": "There are $4$ male athletes and $3$ female athletes.<br/>$(1)$ Now $7$ athletes are lined up. If all female athletes are together, how many ways are there to arrange them?<br/>$(2)$ Now the male athletes are sent to two different venues for training, with at least one athlete in each venue. Each a... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "7577",
"split": "train"
} |
math | [
{
"content": "Determine the smallest integer $B$ such that there exists a sequence of consecutive integers, which must include at least one odd and one even number, adding up to 2023. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1011",
"style": "rule"
} | {
"index": "7578",
"split": "train"
} |
math | [
{
"content": "Evaluate the infinite geometric series: $$\\frac{4}{3} - \\frac{5}{12} + \\frac{25}{144} - \\frac{125}{1728} + \\dots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{64}{63}",
"style": "rule"
} | {
"index": "7579",
"split": "train"
} |
math | [
{
"content": "Find the ordered pair $(m,n)$, where $m,n$ are positive integers satisfying the following equation:\n$$14mn = 55 - 7m - 2n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1,3)",
"style": "rule"
} | {
"index": "7580",
"split": "train"
} |
math | [
{
"content": "Given the random variable $\\xi$ follows a binomial distribution $B(5,0.5)$, and $\\eta=5\\xi$, calculate the values of $E\\eta$ and $D\\eta$ respectively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{125}{4}",
"style": "rule"
} | {
"index": "7581",
"split": "train"
} |
math | [
{
"content": "Given $\\tan \\alpha = \\frac{1}{3}$, calculate the value of $\\frac{\\cos^2 \\alpha - 2\\sin^2 \\alpha}{\\cos^2 \\alpha}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{9}",
"style": "rule"
} | {
"index": "7582",
"split": "train"
} |
math | [
{
"content": "Find the polynomial of minimal degree, in $x,$ which has rational coefficients, leading coefficient $1$, and roots $1+\\sqrt{2}$ and $1+\\sqrt{3}.$ (Write the terms in decreasing order of degree.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x^4-4x^3+x^2+6x+2",
"style": "rule"
} | {
"index": "7583",
"split": "train"
} |
math | [
{
"content": "The expression \\(\\frac{3}{10}+\\frac{3}{100}+\\frac{3}{1000}\\) can be simplified by converting each fraction to a decimal, and then calculating the sum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.333",
"style": "rule"
} | {
"index": "7584",
"split": "train"
} |
math | [
{
"content": "What is the least common multiple of the numbers 1332 and 888? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2664",
"style": "rule"
} | {
"index": "7585",
"split": "train"
} |
math | [
{
"content": "There are 101 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "7586",
"split": "train"
} |
math | [
{
"content": "On a circle, points $A, B, C, D, E, F, G$ are located clockwise as shown in the diagram. It is known that $AE$ is the diameter of the circle. Additionally, it is known that $\\angle ABF = 81^\\circ$ and $\\angle EDG = 76^\\circ$. How many degrees is the angle $FCG$? Please output the final answer ... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "7587",
"split": "train"
} |
math | [
{
"content": "Find the limit of the function:\n\n\\[\n\\lim _{x \\rightarrow 1}\\left(\\frac{x+1}{2 x}\\right)^{\\frac{\\ln (x+2)}{\\ln (2-x)}}\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "7588",
"split": "train"
} |
math | [
{
"content": "Alli rolls a standard $6$-sided die twice. What is the probability of rolling integers that differ by $2$ on her first two rolls? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{2}{9}",
"style": "rule"
} | {
"index": "7589",
"split": "train"
} |
math | [
{
"content": "William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per h... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "7590",
"split": "train"
} |
math | [
{
"content": "A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\\frac{14}{25}$. Find ... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "7591",
"split": "train"
} |
math | [
{
"content": "The walls of a room are in the shape of a triangle $A B C$ with $\\angle A B C=90^{\\circ}, \\angle B A C=60^{\\circ}$, and $A B=6$. Chong stands at the midpoint of $B C$ and rolls a ball toward $A B$. Suppose that the ball bounces off $A B$, then $A C$, then returns exactly to Chong. Find the len... | {
"ground_truth": "3\\sqrt{21}",
"style": "rule"
} | {
"index": "7592",
"split": "train"
} |
math | [
{
"content": "Find all polynomials of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\\dots+a_1x+(-1)^n(n+1)$$ with integer coefficients, having $n$ real roots $x_1,\\dots,x_n$ satisfying $k \\leq x_k \\leq k+1$ for $k=1, \\dots,n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "P_1(x) = x - 2",
"style": "rule"
} | {
"index": "7593",
"split": "train"
} |
math | [
{
"content": "Calculate: \\(\\left[\\left(11 \\frac{1}{9}-3 \\frac{2}{5} \\times 1 \\frac{2}{17}\\right)-8 \\frac{2}{5} \\div 3.6\\right] \\div 2 \\frac{6}{25}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20}{9}",
"style": "rule"
} | {
"index": "7594",
"split": "train"
} |
math | [
{
"content": "For a certain positive integer $n,$ there exist real numbers $x_1,$ $x_2,$ $\\dots,$ $x_n$ such that\n\\begin{align*}\nx_1 + x_2 + x_3 + \\dots + x_n &= 1000, \\\\\nx_1^4 + x_2^4 + x_3^4 + \\dots + x_n^4 &= 512000.\n\\end{align*}Find the smallest positive integer $n$ for which this is possible. Pl... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "7595",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $c=a\\cos B+b\\sin A$. \n(I) Find angle $A$. \n(II) If $a=2$, find the maximum area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "\\sqrt {2}+1",
"style": "rule"
} | {
"index": "7596",
"split": "train"
} |
math | [
{
"content": "There are 21 competitors with distinct skill levels numbered $1,2, \\ldots, 21$. They participate in a pingpong tournament as follows. First, a random competitor is chosen to be \"active\", while the rest are \"inactive.\" Every round, a random inactive competitor is chosen to play against the cur... | {
"ground_truth": "\\frac{47}{42}",
"style": "rule"
} | {
"index": "7597",
"split": "train"
} |
math | [
{
"content": "Divide every natural number with at least two digits by the sum of its digits! When will the quotient be the largest, and when will it be the smallest? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.9",
"style": "rule"
} | {
"index": "7598",
"split": "train"
} |
math | [
{
"content": "It is very boring to look at a black-and-white clock face, so Clive painted the number 12 red exactly at noon and decided to paint the current hour red every 57 hours.\n\na) How many numbers on the clock face will be painted?\n\nb) How many numbers will be red if Clive paints them every 1913th hou... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7599",
"split": "train"
} |
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